On a conjecture of Gowers and Wolf

TL;DR

This paper proves a stronger version of Gowers and Wolf's conjecture, showing that certain multilinear averages can be controlled by Gowers norms based on linear independence conditions of associated functions.

Contribution

The paper confirms and strengthens the conjecture by Gowers and Wolf, establishing control of multilinear averages via Gowers norms under linear independence assumptions.

Findings

Confirmed the Gowers-Wolf conjecture.

Established control of averages by Gowers norms.

Extended the conjecture to a stronger version.

Abstract

Gowers and Wolf have conjectured that, given a set of linear forms $\{\psi_i\}_{i=1}^t$ each mapping $\mathbb{Z}^D$ to $\mathbb{Z}$, if $s$ is an integer such that the functions $\psi_1^{s+1},\ldots, \psi_t^{s+1}$ are linearly independent, then averages of the form $\mathbb{E}_{\boldsymbol{x}} \prod_{i=1}^t f(\psi_i(\boldsymbol{x}))$ may be controlled by the Gowers $U^{s+1}$-norm of $f$. We prove (a stronger version of) this conjecture.